JOURNAL OF COMPUTATIONAL PHYSICS
ARTICLE NO.
138, 251–285 (1997)
CP975454
High-Order Accurate Discontinuous Finite
Element Solution of the 2D Euler Equations
F. Bassi* and S. Rebay†
*Dipartimento di Energetica, Università degli Studi di Ancona, Via Brecce Bianche, 60131 Ancona,
Italy; †Dipartimento di Ingegneria Meccanica, Università degli Studi di Brescia,
Via Branze 38, 25121 Brescia, Italy
E-mail: [email protected]
Received August 1, 1995; revised February 23, 1996
This paper deals with a high-order accurate discontinuous finite element
method for the numerical solution of the Euler equations. The method combines two key ideas which are at the basis of the finite volume and of the
finite element method, the physics of wave propagation being accounted for
by means of Riemann problems and accuracy being obtained by means of
high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular we show that, in the presence of curved
boundaries, a meaningful high-order accurate solution can be obtained only
if a corresponding high-order approximation of the geometry is employed.
We present numerical solutions of classical test cases computed with linear,
quadratic, and cubic elements which illustrate the versatility of the method
and the importance of the boundary condition treatment. Q 1997 Academic Press
Key Words: Euler equations; discontinuous Galerkin; higher order accuracy; boundary conditions.
1. INTRODUCTION
This paper deals with a high-order accurate discontinuous finite element method
for the numerical solution of the Euler equations on unstructured grids. The method
is based on a discontinuous finite element space discretization originally considered
by Lesaint and Raviart [16, 17] for the solution of the neutron transport equation
and more recently extended by Cockburn et al. [9–12] for the solution of nonlinear
systems of conservation laws. The method combines two key ideas characterizing
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Copyright  1997 by Academic Press
All rights of reproduction in any form reserved.
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BASSI AND REBAY
finite element and (upwind) finite volume methods. As in classical (continuous) finite
element methods, in fact, accuracy is obtained by means of high-order polynomial
approximation within elements rather than by means of wide stencils as in finite
volume schemes. As a consequence, there is no need for the interpolation procedures
used in finite volume methods which, especially for unstructured meshes (see, e.g.,
Barth [2], Harten and Chakravarthy [15]), are not at all a trivial task. At the same
time, the discontinuous approximation of the solution leads naturally to a treatment
of the numerical flux at element interfaces based on the solution of Riemann
problems, thus introducing a proper upwinding in the method which, in this respect,
is similar to a finite volume scheme. The method is therefore particularly well suited
to compute high-order accurate solutions of hyperbolic problems on unstructured grids.
Besides the upwind treatment of the fluxes at element interfaces, high-order
discontinuous finite element methods require some additional nonlinear dissipation
in order to capture the physically relevant weak solutions of the Euler equations
without oscillations. A convenient dissipative mechanism is the local projection
limiting procedure introduced by Cockburn and Shu in the series of papers [9–12].
Most of the published works on the discontinuous finite element method (see,
e.g., Cockburn and Shu, Bey and Oden [7], and the recent work of Biswas, Devine,
and Flaherty [8] on an h-p adaptive discontinuous finite element) concentrate on
transient problems on polygonal domains and make use of quadratic polynomials
at most. This work is in some sense complementary to the previous ones in that
we focus our attention to steady-state subsonic and shockless transonic problems
which are computed without any form of limiting and in that we consider domains
with curved boundaries. In particular we show the feasibility and the convenience
of highly accurate (up to fourth order) steady-state Euler computations on relatively
coarse unstructured grids of triangles. Second, and more important, we put in
evidence that the method is inaccurate at curved solid walls if a piecewise linear
approximation of the geometry of the boundary is employed and show that a higher
order boundary representation introduces a dramatic improvement in the accuracy
of the numerical approximation.
Notice that the accuracy of most finite volume schemes is likewise strongly
affected by the treatment of the solid wall slip condition. Over the years, several
procedures for prescribing the slip condition have appeared in the literature. Solid
wall boundary treatments of improved accuracy (see, for example, the procedure
suggested in [13]) show that most of the extrapolation techniques commonly used
in cell-centered finite volume methods lead to numerical solutions which are quite
less accurate at boundary points than at internal points. However, the discontinuous
finite element method requires a more accurate treatment of the slip condition than
that allowed by common finite volume methods. In fact we are going to show that
it is mandatory to represent accurately the real (curved) geometry of the boundary
in order to compute meaningful solutions that otherwise cannot be obtained even
on extremely refined grids.
In the section of computational results we present accurate (up to fourth-order)
two-dimensional subsonic and shockless transonic computations that emphasize
both the versatility of the proposed method and the influence of the solid wall
boundary treatment on the solution accuracy.
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SOLUTION OF 2D EULER EQUATIONS
2. DISCONTINUOUS GALERKIN FORMULATION
We consider the Euler equations written in conservation form,
­u
1 = ? F(u) 5 0,
­t
(1)
equipped with suitable initial-boundary conditions. The conservative variables u
and the cartesian components f(u) and g(u) of the flux function F(u) are given by
56 5 6 5 6
r
u5
ru
ru
; f(u) 5
rv
rv
ruu 1 p
re
; g(u) 5
rvu
ruv
rvv 1 p
rhu
,
(2)
rhv
where r is the fluid density, u and v are the velocity components, p is the pressure,
and e is the total internal energy per unit mass. The total enthalpy per unit mass
h is defined as h 5 e 1 p/ r. By assuming that the fluid obeys the perfect gas state
equation, p can be calculated as p 5 (c 2 1)r(e 2 (u2 1 v2)/2), where c indicates
the ratio between the specific heats of the fluid.
By multiplying by a ‘‘test function’’ v, integrating over the domain V, and performing an integration by parts we obtain the weak statement of the problem
E
V
v
­u
dV 1
­t
R
­V
vF(u) ? n ds 2
E
V
=v ? F(u) dV 5 0 ;v,
(3)
where ­V denotes the boundary of V.
A discrete analogue of Eq. (3) is obtained by subdividing the domain V into a
collection of nonoverlapping elements hE j and by considering functions uh and vh ,
defined within each element, given by the combination of n shape functions fi ,
uh (x, t) 5
O U (t)f (x),
n
i
i51
i
vh (x) 5
O V f (x)
n
i
i51
i
;x [ E.
(4)
The expansion coefficients Ui (t) and Vi denote the degrees of freedom of the
numerical solution and of the test function for an element E. The definition of the
shape functions f and of the geometry of the elements hE j will be given in the
following. Notice, however, that there is no global continuity requirement for uh
and vh , which are therefore discontinuous functions across element interfaces.
By splitting the integrals over V appearing in Eq. (4) into the sum of integrals
over the elements E and by admitting only the functions uh and vh , we obtain the
semidiscrete equations for a generic element E, which can be written as
d
dt
E
E
vhuh dV 1
R
­E
vh F(uh) ? n ds 2
E
E
where ­E denotes the boundary of element E.
=vh ? F(uh) dV 5 0 ;vh ,
(5)
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BASSI AND REBAY
Equation (5) must be satisfied for any element E and for any function vh . However,
within each element, the vh are a linear combination of n shape functions fj , and
Eq. (5) is, therefore, equivalent to the system of n equations
d
dt
E
E
fj uh dV 1
R
­E
fj F(uh) ? n ds 2
E
E
=fj ? F(uh) dV 5 0, 1 # j # n.
(6)
The Galerkin mechanism therefore constructs automatically a system of n equations
which is sufficient to determine the n degrees of freedom Ui (t) of the unknown
solution uh .
Due to the discontinuous function approximation, flux terms are not uniquely
defined at element interfaces. It is at this stage that the technique traditionally used
in finite volume schemes is borrowed by the discontinuous finite element method.
The flux function F(uh) ? n appearing in the second term of Eq. (5) is in fact replaced
by a numerical flux function h(uh2, uh1) which depends on the internal interface state
uh2, on the neighbouring element interface state uh1, and on the direction n normal
to the interface. The states uh2 and uh1 are calculated by means of the expansion
appearing in the first relation of Eq. (4) evaluated at the element interfaces. In
order to guarantee the formal accuracy of the scheme, h(u2, u1) is required to
satisfy the relations
h(u, u) 5 F(u) ? n, h(u, v) 5 2h(v, u),
(7)
which are called the consistency condition and the directional consistency condition,
respectively. There are several numerical flux functions satisfying the above criteria
such as the ‘‘exact’’ Riemann flux function or the approximate Lax–Friedrichs,
Roe, Engquist–Osher, or Harten–Lax–van Leer (HLLE) flux functions. In this
work all the computations have been performed with the Riemann flux.
Notice that Eq. (5), with the flux function F(uh) ? n, replaced by the numerical
flux function h(uh2, uh1), is nothing but the standard Galerkin method applied to just
one element in which the weakly imposed boundary data are obtained from the
neighbouring elements. Considering, for example, the Roe flux function
h(uh2, uh1) 5 As [F(uh1) 1 F(uh2)] ? n 2 As uAu(uh1 2 uh2) 5 F(uh2) ? n 1 A2(uh1 2 uh2),
the equation for the element E can be rewritten as
d
dt
E
E
fj uh dV 1
52
R
­E
R
­E
fj F(uh2) ? n ds 2
E
E
=fj ? F(uh) dV 2
R
­E
fj A2uh2 ds
fj A2uh1 ds,
where A 5 [­F(uhR)/­u] ? n is the Jacobian matrix evaluated at the Roe average
state uhR, uAu 5 LuLuR, A2 5 LL2R, L and R being the matrices of the left and
right eigenvectors of A, uLu 5 diaghul k uj and L2 5 diaghmin(l k , 0)j being the diagonal
matrices of the absolute and of the negative eigenvalues of A. This amounts to
SOLUTION OF 2D EULER EQUATIONS
255
weakly prescribing the variables carried along the characteristics entering the
element.
Despite the upwind treatment of the fluxes at the element interfaces, the numerical solutions computed with the method described so far (excluding the special case
of a single constant shape function f for each element) are oscillatory in the vicinity
of a flow discontinuity if this is not located exactly at element interfaces. As shown
by Godunov theorem [14], this is a typical behaviour of any second or high-order
accurate numerical scheme in the vicinity of a discontinuous solution. In order to
compute physically relevant solutions without oscillations, it is therefore necessary
to introduce into the discontinuous finite element method some nonlinear dissipative
mechanism which does not destroy the formal order of accuracy of the scheme,
such as the local projection limiting strategy proposed by Cockburn et al. in [11].
Another possibility is to augment the scheme with a shock-capturing term similar
to that commonly considered in the SUPG finite element method (see, e.g., [5]).
In this work, however, due to the fact that we concentrate on shockless problems,
we did not find it necessary to introduce any kind of limiting procedure.
Let us now describe the geometry of the elements and define the shape functions
employed in this work. We have used 2D triangular elements with possibly curved
edges in all the computations. It is common practice in the finite element method
to deal with such curved elements by introducing a reference element Ê in a
nondimensional space and a geometric transformation which maps the reference
element onto the real elements in the physical space. The mapping from the reference element to the real element is a polynomial function P̂ m of order less than or
equal to m defined in the space of the reference element for each independent
variable. Since the mapping must guarantee the geometric continuity between
neighbouring elements, it is commonly expressed in terms of Lagrangian functions
ĉ im and Lagrangian node coordinates x i and is written as
x h ( j) 5
O x ĉ (j)
l
i
m
i
i 50
;j [ Ê,
(8)
where j is the independent variable in the reference space of Ê. Notice that the
Lagrangian nodes of the element edges belonging to the boundary ­V are placed
on the real geometry of the boundary. A curved boundary is therefore approximated
by a set of piecewise polynomial curved segments which individually are C m, but
which globally form a curve that in general is only C 0.
Also the functions uh and fj are defined as polynomial functions on the reference
element. We therefore introduce the functions ûh and f̂j which are polynomials
P̂ k of degree less than or equal to k within the reference element Ê, i.e.
ûh (j, t) 5
O U (t)f̂ (j)
n
i
i 51
k
i
;j [ Ê.
(9)
The functions uh and f ik for a generic real element E are formally obtained as
uh (xh , t) 5 ûh [ j(xh), t], f k (xh) 5 f̂ k [ j(xh)] ;xh [ E,
(10)
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BASSI AND REBAY
where j 5 j(xh) is the inverse of the mapping of Eq. (8). Notice that the functions
uh and fj are not polynomials for a generic element. However, in the case of linear
mapping (i.e., of real elements with straight edges), the functions uh and fj are in
fact polynomials P k also in the physical space.
Equation (10), however, is not required explicitly, since the integrals of Eq. (6) are
evaluated in the space of the reference element by means of numerical quadrature
formulae. As a consequence we only need Eq. (9) to represent the unknowns
and Eq. (8) to compute the determinant of the inverse Jacobian matrix of the
transformation at each quadrature point.
The number of quadrature points used is chosen to integrate exactly on the
reference element polynomials of order 2k. In the case of linear, quadratic, and
cubic shape functions f, we evaluate the volume integrals with Gauss formulae
using three, six, and twelve points and the interface and the boundary integrals
with formulae using two, three, and four points, respectively.
We can distinguish different types of elements by the order k of the shape
functions f̂ k used to represent the unknowns and by the order m of the shape
functions ĉ m of the geometric mapping. The element is called isoparametric if
m 5 k, subparametric if m , k, and superparametric if m . k. In all the computations
we are going to present in the following section we have used the complete polynomial basis of order m or k, both for the unknowns and for the geometric transformations.
By assembling together all the elemental contributions (6), the system of ordinary
differential equations which govern the evolution in time of the discrete solution
can be written as
M
dU
1 R(U) 5 0,
dt
(11)
where M denotes the mass matrix, U is the global vector of the degrees of freedom,
and R(U) is the residual vector. Since the shape functions f u E are nonzero within
element E only, the mass matrix M has a block diagonal structure which couples
the n degrees of freedom of each unknown variable only within element E. As a
consequence M can be very simply inverted considering one element at a time.
The time integration of the semidiscrete system is accomplished by means of a
Runge–Kutta explicit method for initial value problems. In practice we have used
the Runge–Kutta methods advised by Cockburn and Shu in [9], which are linearly
stable for a Courant number less than or equal to 1/(2k 1 1). This rather restrictive
CFL condition might suggest looking for a more efficient integration scheme. Preliminary work in this direction seems to indicate that an implicit method is indeed more
efficient, especially for steady state computations. In the present work, however, this
topic has not been addressed.
3. BOUNDARY TREATMENT
When the boundary ­E of an element E belongs to ­V, the normal flux function
F(uh ) ? n must be consistent with the boundary condition to be imposed on ­V and
SOLUTION OF 2D EULER EQUATIONS
257
will be denoted by hbc . The boundary normal flux function hbc is computed in terms
of the numerical flux function as hbc 5 h(uh2, ubc), where uh2 is the internal boundary
state at the current time level and ubc is chosen according to the conditions that
must be satisfied on the boundary. At inflow/outflow boundaries, the state ubc is
computed by means of a locally one-dimensional characteristic analysis in a direction
normal to the boundary by imposing the available data and the Riemann invariants
associated to outgoing characteristics. On solid walls a symmetry technique is used
whereby the state ubc has the same density, internal energy, and tangential velocity
component of uh2 and the opposite normal velocity component.
The simplest element considered is the isoparametric linear element (k 5 m 5
1). Since the geometric mapping is piecewise linear, the boundary of the domain
is in this case discretized as the polygonal line that joins the grid points. Such a
linear geometric discretization of the boundary gives rise to errors in the numerical
solution which, for the Euler equations in conservation form, appear as spurious
entropy production near solid walls. The usual remedy for this accuracy loss consists
of clustering the grid points in the regions of high curvature. In general, accurate
solutions are found, provided that the mesh is fine enough to represent smoothly
the boundary geometry. However, recent work on boundary conditions for finite
volume methods applied to the Euler equations indicates that, on a given mesh,
much more accurate results are obtained by using a solid wall boundary treatment
that takes into account the wall curvature (see, e.g., [13]).
During earlier attempts to compute the numerical solution of rather standard
subsonic and transonic flows over airfoils [3–5], we have discovered that the discontinuous finite element method suffers from a piecewise linear approximation of the
geometry of solid boundaries much more than standard finite volume schemes. The
computations presented in the following section seem to indicate that a high-order
geometric approximation of curved boundaries is mandatory if accurate numerical
solutions are sought (i.e., we must use m $ 2, ;k). In fact, we have not been able
to compute accurate numerical results using geometrically linear elements even on
highly refined grids.
4. NUMERICAL RESULTS
The discontinuous finite element method has been implemented in a computer
code that can use elements of different geometric shape and high order polynomial
approximations both for the unknowns and for the geometric mapping functions.
Due to the explicit nature of the time stepping scheme, we have chosen an
implementation aimed to minimize the computer time whereby the block diagonal
mass matrix, the determinant of the Jacobian of the coordinate transformation, and
the gradient of the shape functions are stored at the Gauss points of each element
once for all at the beginning of the computation. With this choice, the memory
requirement per element is of 22 integer words and 85 real (double precision) words
for linear elements, of 33 integer words and 216 real words for quadratic elements
and of 46 integer words and 518 real words for cubic elements. However, if the
mass matrix, the Jacobian determinants, and the shape function gradients are recomputed at every iteration, the storage requirement reduces by a factor ranging from
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BASSI AND REBAY
2 for linear elements to 3.5 for cubic elements, while the computer time increases
by about 50% in all cases.
All the computations (which converged to a steady-state solution) have been
marched to a 10210 error in the L2 norm of the residuals and have been performed
using double precision arithmetic on a HP735-125 workstation. The CPU time per
time step (two Runge–Kutta stages) per element is of 0.104, 0.244, and 0.512 ms
for linear, quadratic, and cubic elements, respectively. For the medium density
32 3 8 grid considered in the next section, the computations performed with linear,
quadratic, and cubic elements took 6560, 21559, and 95578 time steps to converge
to a density residual L2 norm equal to 10210.
The results presented in the following sections are computed on very regular
unstructured grids of triangles. Other results [5] show that the quality of the solution
does not significantly reduce by employing more general triangulations obtained
with mesh generation algorithms which can treat arbitrary domains.
4.1. Flow around a Circle
We first present the results obtained for the 2D subsonic flow around a circle at
Mach number My 5 0.38, a typical test case considered for the evaluation of
numerical methods for the Euler equations.
In order to check the accuracy and the convergence properties of the method, we
have performed various computations using different combinations of interpolation
functions for the unknowns and for the geometric mapping from the reference
element to the real elements. Different elements are denoted by PkQm, where k
indicates the order of the complete polynomials used to approximate the unknowns
and m indicates the order of the complete polynomials used for the geometric
mapping.
We present computations with linear, quadratic, and cubic approximation of the
unknowns and with linear, quadratic, and cubic geometric mapping. The various
computations have been performed on four successively refined grids shown in
Figs. 1–4, which are the Delaunay triangulations obtained from the distributions
of 16 3 4, 32 3 8, 64 3 16, and 128 3 32 points, respectively. All the grids extend
about 20 diameters away from the circle.
4.1.1. Second-Order Accurate Computations
Figures 5–8 show the Mach isolines computed with P1Q1 isoparametric elements
on the four grids. The isolines are plotted for values of the Mach number given by
M i 5 M0 1 i DM, i 5 0, 1, ..., where M0 5 0 and DM 5 0.038. These solutions are
very inaccurate, as put in evidence by the nonphysical ‘‘boundary layer’’ which
develops along the solid wall and by the associated ‘‘wake’’ in the downstream
region of the circle. Moreover, the computations do not converge to a steady
solution because of the unsteadiness of the unphysical wake. The Mach isolines
shown in the figures are therefore those at some instant in time which corresponds
to 100,000 Runge–Kutta cycles. Notice that the poor behaviour of the numerical
solution at the boundary does not disappear even on the very refined 128 3 32 grid.
SOLUTION OF 2D EULER EQUATIONS
FIG. 1. 16 3 4 grid around a circle.
FIG. 2. 32 3 8 grid around a circle.
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BASSI AND REBAY
FIG. 3. 64 3 16 grid around a circle.
FIG. 4. 128 3 32 grid around a circle.
SOLUTION OF 2D EULER EQUATIONS
FIG. 5. Mach isolines around a circle with P1Q1 elements on the 16 3 4 grid.
FIG. 6. Mach isolines around a circle with P1Q1 elements on the 32 3 8 grid.
261
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BASSI AND REBAY
FIG. 7. Mach isolines around a circle with P1Q1 elements on the 64 3 16 grid.
FIG. 8. Mach isolines around a circle with P1Q1 elements on the 128 3 32 grid.
SOLUTION OF 2D EULER EQUATIONS
263
FIG. 9. Mach isolines around a circle with P1Q2 elements on the 16 3 4 grid.
The picture changes completely when using P1Q2 superparametric elements, as
reported in Figs. 9–12, which show the Mach isolines computed on the four successively refined grids. In this case the numerical solution converges nicely to a solution
which is (nearly) symmetric in the upstream–downstream direction and which is
characterized by a limited entropy production along the wall, as reported in Table
I which shows the L2 entropy error and the order of convergence of the numerical
solutions in comparison with other higher order ones (to be discussed in the following).
In order to gain some insight into the unexpected poor behaviour at a curved
boundary of the P1Q1 discontinuous Galerkin finite element computations, we
have performed some test calculations with a hybrid element (called P1Q1*) in
which the boundary edges are regarded as straight lines as in Q1 elements but the
normal direction changes locally along the boundary according to the real curved
geometry as in Q2 elements.
Figures 13–16 show the Mach isolines computed with P1Q1* elements. The
solution is more accurate than that computed with P1Q1 elements but remains
much more inaccurate than that computed with P1Q2 elements. We have also
performed other computations (not reported for reasons of brevity) using curved
boundary edges and constant normal direction along each boundary edge, which
resulted in less accurate solutions than those computed with P1Q1* elements. This
means that the accuracy gain obtained by using a high-order representation of the
geometry of the boundary is mainly related to the correct distribution of the normal
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BASSI AND REBAY
FIG. 10. Mach isolines around a circle with P1Q2 elements on the 32 3 8 grid.
FIG. 11. Mach isolines around a circle with P1Q2 elements on the 64 3 16 grid.
265
SOLUTION OF 2D EULER EQUATIONS
FIG. 12. Mach isolines around a circle with P1Q2 elements on the 128 3 32 grid.
direction along curved boundaries. However, our computations seem to indicate
that a quadratic representation of the boundary is mandatory in order to obtain
accurate solutions.
4.1.2. Higher Order Accurate Computations
We next present the result obtained with P2Q2 and P3Q3 isoparametric elements.
Because of the inaccuracy already shown by the P1Q1 computations, no attempt
has been made to use a linear geometric mapping in conjunction with high-order
interpolation functions for the unknowns. The results of additional test calculations
TABLE I
L2 Entropy Errors and Orders of Convergence for the P1Q2, P2Q2, and
P3Q3 Computations of the Flow around a Circle on Four Successive Grids
P1Q2
P2Q2
P3Q3
Erra
Errb
Errc
Errd
Ordab
Ordbc
Ordcd
0.710 3 1021
0.135 3 1021
0.818 3 1022
0.102 3 1021
0.941 3 1023
0.415 3 1023
0.153 3 1022
0.649 3 1024
0.182 3 1024
0.231 3 1023
0.614 3 1025
2.78
3.84
4.30
2.74
3.86
4.51
2.73
3.40
Note. The indices a, b, c, and d indicate the 16 3 4, 32 3 8, 64 3 16, and 128 3 32 grids, respectively.
Ordab , indicates the order of convergence obtained by comparing the solutions of grid a and of grid b
and is computed as Ordab 5 log(Erra /Errb)/log(ha /hb), h being the mesh size of the grid.
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BASSI AND REBAY
FIG. 13. Mach isolines around a circle with P1Q1* elements on the 16 3 4 grid.
FIG. 14. Mach isolines around a circle with P1Q1* elements on the 32 3 8 grid.
SOLUTION OF 2D EULER EQUATIONS
FIG. 15. Mach isolines around a circle with P1Q1* elements on the 64 3 16 grid.
FIG. 16. Mach isolines around a circle with P1Q1* elements on the 128 3 32 grid.
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BASSI AND REBAY
FIG. 17. Mach isolines around a circle with P2Q2 elements on the 16 3 4 grid.
(not reported for brevity) performed with P2Q3 and P3Q4 superparametric elements did not show any significant difference from those obtained with P2Q2 and
P3Q3 elements.
Figures 17–20 show the Mach isolines computed with P2Q2 elements on the four
grids, Figs. 21–24 show the Mach isolines computed with P3Q3 elements, and Table
I reports the L2 entropy error and the order of accuracy for the P1Q2, P2Q2, and
P3Q3 computations (the result obtained for P1Q1 elements are not reported, since
these computations do not converge to a steady-state solution).
These computations allow us to appreciate the potentialities offered by thirdand fourth-order accurate elements in the numerical solutions of the Euler equations
on unstructured grids. Notice, in fact, the remarkably small entropy error of these
numerical solutions even on the very coarse 16 3 4 grid. Notice also that the order
of convergence of these computations is consistent with the theoretical one.
The accuracy of the method is also evidenced in Figs. 25–28, which report the
distribution of the pressure coefficient around the circle for the four different grids,
and by Figs. 29–32, which report the distribution of the total pressure loss coefficient
(defined as the ratio between the total pressure at a point and the total pressure
of the undisturbed flow) for the various grids.
4.2. Ringleb Flow
We next present the result obtained for the more difficult Ringleb flow test case,
which corresponds to an analytical smooth solution of the Euler equations obtained
with the ‘‘hodograph method’’ (see, e.g., [1]). The Ringleb solution represents a
SOLUTION OF 2D EULER EQUATIONS
FIG. 18. Mach isolines around a circle with P2Q2 elements on the 32 3 8 grid.
FIG. 19. Mach isolines around a circle with P2Q2 elements on the 64 3 16 grid.
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BASSI AND REBAY
FIG. 20. Mach isolines around a circle with P2Q2 elements on the 128 3 32 grid.
FIG. 21. Mach isolines around a circle with P3Q3 elements on the 16 3 4 grid.
SOLUTION OF 2D EULER EQUATIONS
FIG. 22. Mach isolines around a circle with P3Q3 elements on the 32 3 8 grid.
FIG. 23. Mach isolines around a circle with P3Q3 elements on the 64 3 16 grid.
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BASSI AND REBAY
FIG. 24. Mach isolines around a circle with P3Q3 elements on the 128 3 32 grid.
FIG. 25. Pressure coefficient distribution around a circle with P1Q1* elements.
SOLUTION OF 2D EULER EQUATIONS
FIG. 26. Pressure coefficient distribution around a circle with P1Q2 elements.
FIG. 27. Pressure coefficient distribution around a circle with P2Q2 elements.
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BASSI AND REBAY
FIG. 28. Pressure coefficient distribution around a circle with P3Q3 elements.
FIG. 29. Total pressure loss coefficient distribution around a circle with P1Q1* elements.
SOLUTION OF 2D EULER EQUATIONS
FIG. 30. Total pressure loss coefficient distribution around a circle with P1Q2 elements.
FIG. 31. Total pressure loss coefficient distribution around a circle with P2Q2 elements.
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BASSI AND REBAY
FIG. 32. Total pressure loss coefficient distribution around a circle with P3Q3 elements.
flow which turns around a symmetric obstacle. The flowfield is irrotational and
isentropic, and the streamlines are symmetric along the axis of symmetry of the
obstacle. Their shape tends to become parabolic far away from the body. The flow
field is supersonic in a region around the nose of the obstacle.
Also in this test case we have performed various computations with different
types of elements in order to check the convergence properties and the accuracy
of the method at the solid wall. The elements are denoted as in the circle computations. We present calculations with linear, quadratic, and cubic approximation of
the unknowns and with linear, quadratic, cubic, and fourth-order geometric mapping. The various computations have been performed on the four successively
refined Delaunay grids shown in Figs. 33–36, which are the triangulations of sets
of 16 3 4, 32 3 8, 64 3 16, and of 128 3 32 points.
Figures 37 and 38 show the Mach isolines computed with P1Q1 isoparametric
elements on the two finer grids, while Figs. 39 and 40 show a detail around the
nose of the obstacle of the Mach isolines on the same grids. The isolines are plotted
for values of the Mach number given by M i 5 M0 1 i DM, i 5 0, 1, ..., where
M0 5 0 and DM 5 0.05. Similarly to the circle test case, the numerical solutions
are spoiled by a large spurious entropy production at solid walls if a linear approximation of the geometry is used. Even the solution obtained on the finest grid is very
inaccurate near the solid walls. As shown in Figs. 41–44, however, this inaccuracy is
completely cured by considering P1Q2 (or higher order) elements.
The accuracy of the high-order computations is evidenced in Figs. 45–52, showing
the Mach isolines computed with P2Q3 and P3Q4 elements, and in Table II, which
reports the L2 entropy error and the order of convergence for various types of
element. Similar results are obtained with P2Q2 and P3Q3 isoparametric elements.
SOLUTION OF 2D EULER EQUATIONS
FIG. 33. 16 3 4 grid for the Ringleb flow.
FIG. 34. 32 3 8 grid for the Ringleb flow.
FIG. 35. 64 3 16 grid for the Ringleb flow.
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BASSI AND REBAY
FIG. 36. 128 3 32 grid for the Ringleb flow.
FIG. 37. Ringleb flow Mach isolines with P1Q1 elements on the 64 3 16 grid.
FIG. 38. Ringleb flow Mach isolines with P1Q1 elements on the 128 3 32 grid.
SOLUTION OF 2D EULER EQUATIONS
FIG. 39. Detail of the Ringleb flow Mach isolines on the 64 3 16 grid.
FIG. 40. Detail of the Ringleb flow Mach isolines on the 128 3 32 grid.
FIG. 41. Ringleb flow Mach isolines with P1Q2 elements on the 16 3 4 grid.
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BASSI AND REBAY
FIG. 42. Ringleb flow Mach isolines with P1Q2 elements on the 32 3 8 grid.
FIG. 43. Ringleb flow Mach isolines with P1Q2 elements on the 64 3 16 grid.
FIG. 44. Ringleb flow Mach isolines with P1Q2 elements on the 128 3 32 grid.
281
SOLUTION OF 2D EULER EQUATIONS
TABLE II
L2 Entropy Errors and Orders of Convergence for the P1Q2, P2Q2, P2Q3, P3Q3,
and P3Q4 Computations of the Ringleb Flow on Four Successively Refined Grids
Erra
P1Q2
P2Q2
P2Q3
P3Q3
P3Q4
0.355
0.474
0.611
0.156
0.154
3
3
3
3
3
Errb
1021
1022
1022
1022
1022
0.806
0.803
0.105
0.914
0.970
3
3
3
3
3
Errc
1022
1023
1023
1024
1024
0.167
0.107
0.156
0.624
0.616
3
3
3
3
3
1022
1023
1023
1025
1025
Errd
Ordab
Ordbc
Ordcd
0.395 3 1023
0.150 3 1024
0.222 3 1024
2.14
2.56
2.54
4.09
3.99
2.27
2.91
2.75
3.87
3.98
2.07
2.83
2.81
0.438 3 1026
3.81
Note. The indices a, b, c, and d indicate the 16 3 4, 32 3 8, 64 3 16, and 128 3 32 grids, respectively.
Ordab , indicates the order of convergence obtained by comparing the solutions of grid a and of grid b
and is computed as Ordab 5 log(Erra /Errb)/log(ha /hb), h being the mesh size of the grid.
FIG. 45. Ringleb flow Mach isolines with P2Q3 elements on the 16 3 4 grid.
FIG. 46. Ringleb flow Mach isolines with P2Q3 elements on the 32 3 8 grid.
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BASSI AND REBAY
FIG. 47. Ringleb flow Mach isolines with P2Q3 elements on the 64 3 16 grid.
FIG. 48. Ringleb flow Mach isolines with P2Q3 elements on the 128 3 32 grid.
FIG. 49. Ringleb flow Mach isolines with P3Q4 elements on the 16 3 4 grid.
SOLUTION OF 2D EULER EQUATIONS
FIG. 50. Ringleb flow Mach isolines with P3Q4 elements on the 32 3 8 grid.
FIG. 51. Ringleb flow Mach isolines with P3Q4 elements on the 64 3 16 grid.
FIG. 52. Ringleb flow Mach isolines with P3Q4 elements on the 128 3 32 grid.
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BASSI AND REBAY
5. CONCLUSIONS
A high-order accurate discontinuous finite element method for the solution of
the Euler equations has been presented. The method combines the ability of all
finite element methods of constructing in an automatic and reliable way high-order
approximations of differential problems on general unstructured meshes with the
flux upwinding technique characterizing a class of finite-volume schemes. The
method has proven to be very effective in the solution of classical two-dimensional
steady-state test cases for the Euler equations. Our results show that accurate
solutions on relatively coarse meshes can be computed by using a high-order representation of the unknowns and of the geometry of the boundary of the domain. In
particular, our computations show that a geometric representation which takes into
account (at least) the curvature of the boundary (i.e., a quadratic representation)
is mandatory in order to obtain meaningful numerical solutions.
Work is in progress to extend the method to treat the full Navier–Stokes equations
by resorting to a mixed formulation to discretize the viscous terms. Some preliminary
results obtained with this approach have already been published in [6]. We plan to
give a detailed report about the extension of the method to viscous flows in a
forthcoming paper.
REFERENCES
1. Test Cases for Inviscid Flow Field Methods, AGARD-AR-211, 7 Ancelle 92200, Neuilly sur Seine,
France, 1985.
2. T. J. Barth and P. O. Frederickson, Higher Order Solution of the Euler Equations on Unstructured
Grids Using Quadratic Reconstruction, AIAA-90-0013, 1990.
3. F. Bassi, S. Rebay, and M. Savini, A high resolution discontinuous Galerkin method for hyperbolic
problems on unstructured grids, Numerical Methods in Fluid Dynamics, edited by M. J. Baines and
K. W. Morton, in III-rd ICFD Conference, Reading, April 1992, (Clarendon Press, Oxford, 1993),
p. 345.
4. F. Bassi, S. Rebay, and M. Savini, Discontinuous finite element Euler solutions on unstructured
adaptive grids, Lecture Notes in Physics, Vol. 414, edited by M. Napolitano and F. Sabetta, in XIIIth
ICNMFD, Rome, July 6–10, 1992, (Springer-Verlag, New York/Berlin, 1993), p. 245.
5. F. Bassi and S. Rebay, Accurate 2D Euler computations by means of a high order discontinuous
finite element method, Lecture Notes in Physics, in XIVth ICNMFD, Bangalore, July 11–15, 1994
(Springer-Verlag, New York/Berlin, 1996).
6. F. Bassi and S. Rebay (1995). Discontinuous finite element high order accurate numerical solution
of the compressible Navier–Stokes equations, Numerical Methods in Fluid Dynamics, edited by
M. J. Baines and K. W. Morton in IVth ICFD Conference, Oxford, April 3–6, 1995, (Clarendon
Press, Oxford, UK).
7. K. S. Bey and J. T. Oden, A Runge–Kutta Discontinuous Finite Element Method for High Speed
Flows, AIAA Paper 91-1575-CP, 1991, p. 541.
8. R. Biswas, K. D. Devine, and J. E. Flaherty, Parallel, adaptive finite element methods for conservation
laws, Appl. Numer. Math. 14, 255 (1994).
9. B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite
element method for conservation laws II: General framework, Math. Comput. 52, 411 (1989).
10. B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite
element method for conservation laws III: One dimensional systems, J. Comput. Phys. 84, 90 (1989).
SOLUTION OF 2D EULER EQUATIONS
285
11. B. Cockburn, S. Hou, and C.-W. Shu, The Runge–Kutta local projection discontinuous Galerkin
finite element method for conservation laws IV: The multidimensional case, Math. Comput. 54,
545 (1990).
12. B. Cockburn and C.-W. Shu, ‘‘The P 1 –RKDG Method for two-dimensional Euler Equations of Gas
Dynamics,’’ ICASE Report No. 91-32, 1991.
13. A. Dadone and B. Grossman (1994). ‘‘Surface Boundary Conditions for the Numerical Solution of
the Euler Equations,’’ AIAA Journal, Vol. 32, No. 2, 285–293.
14. S. K. Godunov, ‘‘A Difference Scheme for Numerical Computation of Discontinuous Solutions of
Hydrodynamic Equations’’ Math Sb. 47, 271 (1959). [Russian]. [Translated by U.S. Joint Publ. Res.
Service, JPRS, 7226 (1969).
15. A. Harten and S. R. Chakravarthy, ‘‘Multi-Dimensional ENO Schemes for General Geometries,’’
ICASE Report No. 91-76, 1991.
16. P. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d’éléments finis, Ph.D. thesis, Université Paris 6, 1975.
17. P. Lesaint and P. A. Raviart, On a finite element method to solve the neutron transport equation,
in Mathematical Aspects of Finite Elements in Partial Differential Equations, edited by C. de Boor,
(Academic Press, New York, 1974), p. 89.